Marseille workshop on loops and spin foams

[marcus]

how is the discrete spacetime connected in a topological sense.

I don't think a discrete spacetime model needs to be topologically connected.

At least I never heard it said that one needed a space to be connected before one could define fields and waves and stuff on it.

A lattice of points isn't connected but you can define stuff on it
that looks and acts like waves.

Computers do that all the time, like waves in computer animations are defined on a finite array of points.

Mike2 maybe you are asking the wrong question. Instead of how is the underlying space connected
maybe you should be asking whether and whether there is any reason it needs to be.

I just took a bath in a deep tub of hot water. it looked continuous to me, the water. It acted continuous and connected. It conducts heat and sound and water-waves and is transparent to lightwaves. It would conduct electricity if I was unlucky enough to be struck by lightning while in the bathtub
But it wasnt topologically connected or anythinglike a differentiable manifold.

It was actually a finite set of molecules, behaving like a continuum.

[edit: clarification, I infer from your next post that you thought I was making a reference to LQG, but without mentioning LQG, when i was mentioned lattices! As far as i know LQG is not a lattice theory and does not model space by discrete points. It has an underlying manifold, just no pre-specified geometry. We really need a general classifier word for
the various background indep. approaches to quantizing General Relativity.
they have a lot of resemblances but differ in details. What shall we call them, maybe "Loop etc. gravity" so that it is clear we are including the spin foam and simplicial models?]

 

[Mike2]

I don't think a discrete spacetime model needs to be topologically connected.

At least I never heard it said that one needed a space to be connected before one could define fields and waves and stuff on it.

A lattice of points isn't connected but you can define stuff on it
that looks and acts like waves.

Computers do that all the time, like waves in computer animations are defined on a finite array of points.

Mike2 maybe you are asking the wrong question. Instead of how is the underlying space connected
maybe you should be asking whether and whether there is any reason it needs to be.

I just took a bath in a deep tub of hot water. it looked continuous to me, the water. It acted continuous and connected. It conducts heat and sound and water-waves and is transparent to lightwaves. It would conduct electricity if I was unlucky enough to be struck by lightning while in the bathtub
But it wasnt topologically connected or anythinglike a differentiable manifold.

It was actually a finite set of molecules, behaving like a continuum.

It seems necessary to me that there be some sort of connected space or connected items in order to transmit any kind of signal through a medium. Otherwise, how does information travel from one point to the next if there is absolutely no medium of any kind between the points? So I wonder how the lattice of LQG is connected. Perhaps information travels through the edges. But then can information travel through an infinitesimally thin line? Is LQG creating point particles of space-time, with action at a distance through no medium at all? What?

 

[marcus]

... So I wonder how the lattice of LQG is connected...

Mike2, as selfAdjoint has explained to you in another thread, LQG is based on a continuum, on a differentiable manifold, not on a lattice. You don't have to worry about it being connected.
As sA also remarked the simplicial AJL model, which is really more the topic of this thread, is also a continuum. Think of it as a diff. manif that has been "triangulated" ----built up out of simplices----fused glued welded together from simplices----partitioned into simplices without actually splitting them (they touch).
We need to get on with following developments around the SQG (simplicial quantum gravity) or "dynamical triangulations" model of Ambjorn Jurkiewicz Loll

 

[marcus]

the AJL paper and Simplicial Gravity is a fast moving story so I think we should try to keep up on it
Yesterday Baez posted on SPR---some strong statements about AJL approach in response to Charlie Stromeyer

--------Baez post Sunday, quote----
In article <61773ed7.0405240822.1c7108de@posting.google.com>,
Charlie Stromeyer Jr. <cstromey@hotmail.com> wrote:

>Here are three other reasons to be skeptical of discretized approaches
>to gravity:
>
>1) How are such approaches to be made compatible with vector
>supersymmetry (or vsusy) which is a topological type of symmetry that
>appears in both gravity and topological gauge theories [1].

This "vector supersymmetry" is a mathematical feature of certain
field theories - not something that anyone has observed experimentally.

Nobody has yet constructed a background-free quantum theory that has
general relativity as its limit at large distance scales. The Ambjorn-
Jurkiewicz-Loll model is the closest anyone has come. If they succeed,
this will be of interest regardless of whether their model displays
mathematical features that appear in certain other theories!

>2) How are such approaches to be made compatible with Bell-like
>correlations, non-locality and non-causality which are each present in
>the experiment described in this brief four page paper [2].

As a quantum theory, the Ambjorn-Jurkiewicz-Loll model automatically
has Bell-like "entanglement" and all that jazz.

>3) To paraphrase a sentence that Stephen Hawking once wrote, to not
>believe in the beauty and unity of the dualities of M-theory is like
>believing that evolution did not occur because instead God placed by
>hand all the fossils in the Earth just to play a joke on the
>paleontologists :-)

We resort to theological arguments in physics only when better arguments
are lacking. If a scintilla of experimental evidence for M-theory is
ever found, people will instantly stop making arguments of the sort
you mention here.

Please understand what I'm saying:

I'm not saying that M-theory is "wrong" or that the Ambjorn-Jurkiewicz-Loll
model is "right". M-theory makes too few definite predictions to be wrong.
The AJL model does not include matter, so it cannot be right. But the
AJL model is *interesting*, because it represents the best attempt so far
to find a background-free quantum theory that reduces to general relativity
in the large-scale limit!

---------end quote-------

for me the key point in this post is a mathematician's or mathematical physicist's judgement call:

Nobody has yet constructed a background-free quantum theory that has
general relativity as its limit at large distance scales. The Ambjorn-
Jurkiewicz-Loll model is the closest anyone has come.

 

[marcus]

Another recent Baez post on the AJL paper, this time in response to Thomas Larsson:
-------quote from Sunday 6 June SPR post----

In article <24a23f36.0405170344.69e74067@posting.google.com>,
Thomas Larsson <thomas_larsson_01@hotmail.com> wrote:

>1. Is the AJL model really quantum?

Yes! It has a Hilbert space of states, observables described
as noncommuting self-adjoint operators on this Hilbert space,
and discrete time evolution described by unitary operators on
this Hilbert space.

>Some time ago, Urs
>Schreiber argued that LQG, or at least the LQG string,
>fails to be a true quantum theory, and I tend to agree.

I disagree, but it's not really relevant here: we're not
talking about those other theories.

>However, the AJL model can be viewed as a statistical
>lattice model, and if such a model has a good continuum
>limit, it is AFAIK always described by some kind of QFT.
>What else could it be?

Right!

>2. Is the AJL model really gravity. The action is a rather
>straightforward discretization of the Einstein action with
>a cosmological term:
>
> sum over (d-2)-simplices
>
> det g = volume => sum over d-simplices.
>
>What is perhaps somewhat unusual is that all edges have
>the same length, which is different from Regge calculus.
>Nevertheless, I don't think that this really matters, but
>one could check if the results look different if you
>allow for variable edge lengths.

Right! But, the test of whether the model "is really
gravity" is to carefully examine its behavior in the limit
of large distance scales (i.e. lots of 4-simplices). One
can't easily guess this from looking at the action.
Nonperturbative effects are too important! So, in the
absence of good analytical techniques, one really needs
to run computer simulations - as AJL are doing.

>3. Is the measure right? Here is the place where AJL differ
>significantly from previous simulations. AFAIU, the crux is
>that AJL insist on a strict form of causality: they exclude
>spacetimes where the metric is singular, even at isolated
>points. This may seem like an innoscent restriction, but it
>rules out things like topology change and baby universes,
>which require that the metric be singular somewhere.
>
>It is not obvious to me whether one should insist on such a
>strong form of causality or not, but this assumption leads
>at least to better results, e.g. a reasonably smooth 4D
>spacetime. Thus, I believe that it is a fair chance that
>AJL have indeed succeeded in quantizing gravity.

The issue of the "right measure" is very tricky, so tricky
in fact that I again think the most efficient way to begin
tackling it is to run computer simulations and see if the
AJL model acts like general relativity at large length scales.

>They do so not by assuming a lot of experimentally unconfirmed
>new physics, but rather by strictly implementing the
>time-honored principles of old physics, especially
>causality. That is cool.

Yes! Very cool!

----end quote----

For me, there are two key statements here:

---exerpts---

>1. Is the AJL model really quantum?

Yes! It has a Hilbert space of states, observables described
as noncommuting self-adjoint operators on this Hilbert space,
and discrete time evolution described by unitary operators on
this Hilbert space.

...

>They do so not by assuming a lot of experimentally unconfirmed
>new physics, but rather by strictly implementing the
>time-honored principles of old physics, especially
>causality. That is cool.

Yes! Very cool!

----end exerpts---

the last is again a professional mathematician's judgement call. It may be time to quantize the theory of gravity we all use and to do that in a way
that does not "assume a lot of experimentally unconfirmed new physics".
It looks cool to these guys to get GR quantized by conservatively implementing the tried-and-true established principles. In other words spare us the fairy tales about extra dimensions and just get the job done.

You pay mathematicians in part to make educated guesses about what is cool and not cool, what is interesting and not interesting, and what might work. Part aesthetic and part a kind of laboriously enhanced common sense. I'm listening to both these guy's judgement.

https://www.physicsforums.com/showthread.php?p=227813#post227813

 

[Mike2]

Mike2, as selfAdjoint has explained to you in another thread, LQG is based on a continuum, on a differentiable manifold, not on a lattice. You don't have to worry about it being connected.
As sA also remarked the simplicial AJL model, which is really more the topic of this thread, is also a continuum. Think of it as a diff. manif that has been "triangulated" ----built up out of simplices----fused glued welded together from simplices----partitioned into simplices without actually splitting them (they touch).

You seem to be missing the fundamental delemma. Or perhaps I'm hard of hearing. To quantize gravity IS to quantize the spacetime metric. But quantizing spacetime would of necessity make spacetime discrete and makes impossible any propagation of signals. This is too fundamental of a delemma. What could possibly fix it? So at the moment is seems impossible that you will ever quantize gravity.

 

[selfAdjoint]

Suppose we did have a complete quantization of spacetime. Then we would have interacting spaceons, no doubt exchanging gravitons, and communicating thus across distances. Where's the problem?

 

[Olias]

You seem to be missing the fundamental delemma. Or perhaps I'm hard of hearing. To quantize gravity IS to quantize the spacetime metric. But quantizing spacetime would of necessity make spacetime discrete and makes impossible any propagation of signals. This is too fundamental of a delemma. What could possibly fix it? So at the moment is seems impossible that you will ever quantize gravity.

Spacetime is relevant to 3+1 Dimensionsn and has a metric which gives constant results.

Space-Field, where 'TIME' is detached(exchanged) from the above metric follows certain values in GR, the major factor it is a 2 dimensional arena, and not 3+1, as one 'lose's' the time component in Einsteins field equations, this compactifies and restrains all measures into a non-time dependant arena.

The simplistic overview is that there are only Directional values of motion, all directions are based on 'back-to-back' interactions, like with all Field Equations the action, re-action are similtainious, for every Positive Action there is a corresponding Negative reaction.

 

[sol2]

Spacetime is relevant to 3+1 Dimensionsn and has a metric which gives constant results.

Space-Field, where 'TIME' is detached(exchanged) from the above metric follows certain values in GR, the major factor it is a 2 dimensional arena, and not 3+1, as one 'lose's' the time component in Einsteins field equations, this compactifies and restrains all measures into a non-time dependant arena.

The simplistic overview is that there are only Directional values of motion, all directions are based on 'back-to-back' interactions, like with all Field Equations the action, re-action are similtainious, for every Positive Action there is a corresponding Negative reaction.

When you get to one dimension(string) what happens then

Gr had to be consistently expressed, but it is surrounded, before and after ?

Gravity and electromagnetism are now one( you can't see it but the one is white )?

 

[marcus]

You seem to be missing the fundamental delemma. Or perhaps I'm hard of hearing. To quantize gravity IS to quantize the spacetime metric. But quantizing spacetime would of necessity make spacetime discrete and makes impossible any propagation of signals. This is too fundamental of a delemma. What could possibly fix it? So at the moment is seems impossible that you will ever quantize gravity.

Suppose we did have a complete quantization of spacetime. Then we would have interacting spaceons, no doubt exchanging gravitons, and communicating thus across distances. Where's the problem?

want to try to respond
no time now since i have to go out briefly

will bring in this quote
-------quote from JB post on SPR Sunday 6 June------
Please understand what I'm saying:

I'm not saying that M-theory is "wrong" or that the Ambjorn-Jurkiewicz-Loll
model is "right". M-theory makes too few definite predictions to be wrong.
The AJL model does not include matter, so it cannot be right. But the
AJL model is *interesting*, because it represents the best attempt so far
to find a background-free quantum theory that reduces to general relativity
in the large-scale limit!

---------end quote-------

It is important to realize that quantizing the geometry of a continuum (a manifold) does not necessarily mean to chop up the manifold into little bits.
the manifold can stay continuous and smooth and connected while its
geometry-observables----areas, volumes, angles---become operators on a hilbertspace.

quantization is a way of representing observables, measurements.
it does not necessarily divide everything in sight into discrete quanta.

Mike2 is right in saying that to quantize gravity means to quantize the metric----in that the metric is one common mathematical representation of the geometry. It does not necessarily mean to divide the metric into little bits or force it to have discretized values. Above all it does not mean one necessarily pulverizes space into little bits! I guess that is one possibility (as selfAdjoint suggests) but it is not the necessary outcome.

Going back to the Seventies (and probably earlier) I think what seemed to a lot of people to be an obvious approach to quantizing GR was to have a smooth manifold and take the space of all (smooth) metrics on that manifold and make a hilbertspace which was
L2 functions on that space of geometries. And then you define operators on that hilbert space.
that is, don't think you have to discretize space and don't think you have to discretize the metric. what you want is to have the measurement of geometric properties like areas correspond to operators on a hilbertspace.
and they might turn out to have discrete spectra.

this approach did not work in the Seventies, although later Rovelli and Smolin did get area and volume operators with discrete spectra. by then (the Nineties) they were using the connection, instead of the metric, to represent the geometry.

None of these approaches recognizes a necessity to divide space up into isolated bits.

And the AJL approach which is the focus of this thread does not either.
differential geometers have been triangulating manifolds for ages (over a hundred years I guess) it is a standard thing
and AJL take a manifold---called S³ in their paper---and
triangulate it in a "dynamical" changing way

So Mike2 you are mistaken when you say:
"But quantizing spacetime would of necessity make spacetime discrete and makes impossible any propagation of signals."

It simply isn't true that quantizing spacetime (or more precisely the geometry of spacetime) would make spacetime discrete.

 

[Mike2]

-------quote from JB post on SPR Sunday 6 June------
Please understand what I'm saying:

I'm not saying that M-theory is "wrong" or that the Ambjorn-Jurkiewicz-Loll
model is "right". M-theory makes too few definite predictions to be wrong.
The AJL model does not include matter, so it cannot be right. But the
AJL model is *interesting*, because it represents the best attempt so far
to find a background-free quantum theory that reduces to general relativity
in the large-scale limit!

---------end quote-------

So it sounds like they are saying that GR does not hold up at very, very small distances. Then quantizing gravity is not equivalent to quantizing (discretizing) spacetime itself. Nevertheless,... discrete causality? That is a contradiction of terms. If a change at a point does not even have a start to an effect on a neighbor, then there is no "immediate" reason why it should have any effect at all.

 

[selfAdjoint]

Mike, you continue to equate quantizing to discretizing. You really need to study more about what quantizing really is. States, operators, and uncertainty, superposition and entanglement. Not "separate chunks".

 

[Mike2]

Mike, you continue to equate quantizing to discretizing. You really need to study more about what quantizing really is. States, operators, and uncertainty, superposition and entanglement. Not "separate chunks".

Admittedly, I am not as informed as many in this field. I am trying to develop a better intuition about all this. And I know that QM does not lend itself to any kind of intuition. That said, I have studied sum higher math and physics. And I don't know of any variables/observables that are quantized that do not take on discrete values. What I am trying to understand is how gravity/spacetime can be "quantized" without being made discrete. And if it is discrete, what does that mean. Your response, of course, is no answer to that. Thank you.

 

[setAI]

What I am trying to understand is how gravity/spacetime can be "quantized" without being made discrete.

hint: fractal structures are in a way both discrete and continous- they represent hierarchies of quantized structures which can seem discrete but are really fundamentally continuous- and vice versa!

interestingly enough fractal structures are what always emerge from chaos- and any truly fundamental view of the ontology of Existence itself suggests that the spacetime/forces/energy/matter of a universe must emerge and crystalize out of an "initially" chaotic state-

ultimately you can either have Existence or Non-existence- if you have existence it must be absolute Chaos because if it existed and wasn't random it must have resulted from some more fundamntal ordered process which excluded an infinity of possible forms- you have to "start" with Chaos-

so the ultimate ontology of existence is Chaos> annihilation of equal-opposite interacting structures > remaining structures seeking entropic equilibration [the fundamental origin of Motion itself] crystalizing into a fractal hierarchy > emergence of seemingly discrete matrices/foams/graphs/lattices that emerge as spacetime vacua/branes > particles/forces

um- but don't listen to me- I think I went off topic- sorry for the crazytalk
___________________________

/:set\AI transmedia laboratories

http://setai-transmedia.com

 

[selfAdjoint]

Admittedly, I am not as informed as many in this field. I am trying to develop a better intuition about all this. And I know that QM does not lend itself to any kind of intuition. That said, I have studied sum higher math and physics. And I don't know of any variables/observables that are quantized that do not take on discrete values. What I am trying to understand is how gravity/spacetime can be "quantized" without being made discrete. And if it is discrete, what does that mean. Your response, of course, is no answer to that. Thank you.

Consider a quantized field. The field is continuous, although discrete packets can be exchanged. Remember that a photon is not just a particle; it also manifests as a continuous wave. In basic quantum mechanics the discreteness comes in the measurement or observation. There can be a discrete set of outcomes (eigenvalues) when the Hermitian operator acts on the continuous state function.

 

[Mike2]

Consider a quantized field. The field is continuous, although discrete packets can be exchanged. Remember that a photon is not just a particle; it also manifests as a continuous wave. In basic quantum mechanics the discreteness comes in the measurement or observation. There can be a discrete set of outcomes (eigenvalues) when the Hermitian operator acts on the continuous state function.

It's easy to visualize these things for quantized fields with respect to spacetime variables, the wave function squared tells you the probability of finding the particle at a certain location and time, etc. But I have difficulty imagining what it would even mean to quantize spaetime itself. Is it like the metric is tells you the probability of finding a particle of spacetime? And what happens to the validity of QED and QCD in a world of quantized gravity/spacetime? Wouldn't QED and QCD have to be reformulated with respect to something other than spacetime so that all quantization procedures are with respect to the same variables? If photons, gluons, and gravitons all must interact, then you'd expect their quantization procedure to be based on some commonality; evidently, spacetime/gravity is NOT that commonality. What then is?

 

[marcus]

It's easy to visualize these things for quantized fields with respect to spacetime variables, the wave function squared tells you the probability of finding the particle at a certain location and time, etc.

Mike2, do you mind if I answer----you asked this of selfAdjoint and he can also answer, anyway I only refer to a part of your question (and don't mean to horn in)


an important analogy. think of a very simple space of locations, like the unit interval or the real axis. you say:
"...the wave function squared tells you the probability of finding the particle at a certain location..."

now think of the set of all metrics on some manifold
that is analogous to the unit interval
the set of all possible geometries on this manifold
can be imagined as itself a mathematical space
and wave functions can be defined on it

"...the wave function squared tells you the probability of finding the geometry of the universe in a certain configuration..."

In practice things may be done differently but this gives you
a rough idea of what quantizing the geometry can mean
the wavefunctions are a hilbert space and
then one has operators on that hilbertspace corresponding to
measuring particular observable facts about the quantum state or wavefunction of the geomtry.

but one never totally nails down the geometry, just as one never nails down the position of a particle on the unit interval or the real axis.
does this make it more understandable?

 

[Mike2]

Mike2, do you mind if I answer----you asked this of selfAdjoint and he can also answer, anyway I only refer to a part of your question (and don't mean to horn in)

You should never appologize for contributing to an open forum. That's what it's here for. Just put your 2 cents in, please.

an important analogy. think of a very simple space of locations, like the unit interval or the real axis. you say:
"...the wave function squared tells you the probability of finding the particle at a certain location..."

now think of the set of all metrics on some manifold
that is analogous to the unit interval
the set of all possible geometries on this manifold
can be imagined as itself a mathematical space
and wave functions can be defined on it

"...the wave function squared tells you the probability of finding the geometry of the universe in a certain configuration..."

In practice things may be done differently but this gives you
a rough idea of what quantizing the geometry can mean
the wavefunctions are a hilbert space and
then one has operators on that hilbertspace corresponding to
measuring particular observable facts about the quantum state or wavefunction of the geomtry.

but one never totally nails down the geometry, just as one never nails down the position of a particle on the unit interval or the real axis.
does this make it more understandable?

That's beginning to make sense, thank you. So would our universe then be a particular one of the geometries (a collapsed wave function), or is it always a superposition, and what we see is a classical limit of a type of "geodesic" average?

This all sounds like a 3rd level of quantization. And just as the 2nd level of quantization cannot be used to describe the 1st level (or can it?), the 3 level cannot be considered on par with the results of the 2nd level? Paths cannot be considered the same as particles, and particles cannot be considered the same as geometries, right? How then can the geometries (gravitons?) interact with particles?

 

[Olias]

You should never appologize for contributing to an open forum. That's what it's here for. Just put your 2 cents in, please.




That's beginning to make sense, thank you. So would our universe then be a particular one of the geometries (a collapsed wave function), or is it always a superposition, and what we see is a classical limit of a type of "geodesic" average?

This all sounds like a 3rd level of quantization. And just as the 2nd level of quantization cannot be used to describe the 1st level (or can it?), the 3 level cannot be considered on par with the results of the 2nd level? Paths cannot be considered the same as particles, and particles cannot be considered the same as geometries, right? How then can the geometries (gravitons?) interact with particles?

Mike you want to review this recent paper, it has an interesting angle of relevence:

http://uk.arxiv.org/abs/quant-ph/0406028
http://uk.arxiv.org/abs/quant-ph/0406029


A previous paper: http://uk.arxiv.org/abs/quant-ph/0308101

 

[Mike2]

This all sounds like a 3rd level of quantization. And just as the 2nd level of quantization cannot be used to describe the 1st level (or can it?), the 3 level cannot be considered on par with the results of the 2nd level? Paths cannot be considered the same as particles, and particles cannot be considered the same as geometries, right? How then can the geometries (gravitons?) interact with particles?

What confuses me is that you are treating a graviton as a particle within some background geometry. But it is suppose to represent a quanta of geometry itself. It seems a particle assumes a backgound geometry used to describe its feature such as where and when it is and how big it is and how fast it is vibrating, etc. So how can one possibly describe a particle of "backgound", what non-background measures can be used to describe it? If the graviton is just another mode of vibration of a string, and strings assume a background, then a graviton cannot be a description of that background geometry, and so it does not describe gravity. I need a better picture because this sound like a contradiction. If quantum gravity means quantum spacetime, how do I visualize this? So all of space is a superposition of various quantum geometries? What does that mean? Does that mean that our particular spacetime is just one of the possible states of quantum geometry/spacetime/gravity? Or if there are other observations of a different quanta of spacetime, then how are the boundaries manifest between the different quanta of spacetimes? Thanks.

 

[selfAdjoint]

Does the graviton represent a quantum of geometry? Certainly not in string physics, where it is a spin 2 particle in a "flat" background spacetime, whose interactions mimic Einstein gravity at a certain level of approximation.

If spacetime ever becomes quantized, surely the quanta will not be gravitons. They may emit and absorb gravitons, though, just as the known quanta emit and absorb various bosons.

 

[Mike2]

Does the graviton represent a quantum of geometry? Certainly not in string physics, where it is a spin 2 particle in a "flat" background spacetime, whose interactions mimic Einstein gravity at a certain level of approximation.

So String theory treats gravity like any other force and ignores spacetime warping of Einstein, is that what you are saying?

If spacetime ever becomes quantized, surely the quanta will not be gravitons. They may emit and absorb gravitons, though, just as the known quanta emit and absorb various bosons.

It seems to me that a quanta of geometry cannot interact with a particle any more than particles can interact with paths.

 

[selfAdjoint]

So String theory treats gravity like any other force and ignores spacetime warping of Einstein, is that what you are saying?

That is exactly right. String theory lives in a 26 or 10 dimensional flat Minkowski space, and the graviton simulates Einstein's equations without any space warping. (There are advanced descendents of string theory where the action determines the spacetime, but I don't know how they work out with gravitons).

It seems to me that a quanta of geometry cannot interact with a particle any more than particles can interact with paths.

Sorry, I don't quite see what this means.

 

[Mike2]

That is exactly right. String theory lives in a 26 or 10 dimensional flat Minkowski space, and the graviton simulates Einstein's equations without any space warping. (There are advanced descendents of string theory where the action determines the spacetime, but I don't know how they work out with gravitons).

It would seem impossible for string theory, then, to explain the background it works in, and so it cannot be a TOE. Nor does it seem likely that the flat space of string theory can explain things at the level of such a tiny universe that the dimensions are curled up. So at what level of energy or expansion is string theory supposed to address? Thanks.

 

[selfAdjoint]

It would seem impossible for string theory, then, to explain the background it works in, and so it cannot be a TOE. Nor does it seem likely that the flat space of string theory can explain things at the level of such a tiny universe that the dimensions are curled up. So at what level of energy or expansion is string theory supposed to address? Thanks.

The energy level is close to, but not at, the Planck level. Do pay attention the the caveat I put in my post. There are newer versions of stringy physics that do address the background space question. I just don't know anything about them.

 

[Mike2]

The energy level is close to, but not at, the Planck level. Do pay attention the the caveat I put in my post. There are newer versions of stringy physics that do address the background space question. I just don't know anything about them.

I understand strings are suppose to explain some of the constants in the Standard Model and leave only the string tension and speed of ligh still unexplained. But that's about it, isn't it?

 

[selfAdjoint]

I don't think SST can really explain the constants in the SM. Supersymmetry is supposed to expain some of them (like the generations of quarks) and at least some of the stringy constructions have low energy forms that look something like supersymmetrical extensions of SM, but that's as close as it gets.

 

[marcus]

I just got back from the Marseille conference on loop quantum gravity and spin foams:

http://w3.lpm.univ-montp2.fr/~philippe/quantumgravitywebsite/

It was really great, so I devoted "week206" of my column This Week's Finds entirely to this conference:

http://math.ucr.edu/home/baez/week206.html

In particular, I spend a lot of time giving a very simple nontechnical introduction to the recent work of Ambjorn, Jurkiewicz and Loll in which they seem to get a 4d spacetime to emerge from a discrete quantum model - something that nobody had succeeded in doing before!

http://www.arXiv.org/abs/hep-th/0404156

I hope this lays to rest certain rumors here that I'd burnt out on quantum gravity.

I want to use the may conference as a window on the important developments that have happened in the first half of 2004 in Quantum Gravity.

there are some papers in the May lineup to notice and also the informal message we got about Lee Smolin's interest in what I think Moffat would call a "Nonsymmetric Gravitational" theory or NGT---a modification of GR's lowenergy Newtonian limit. John Baez referred to it as "MOND" but I think what they were really talking about is the latest version of a mondic-type thing that isn't the crude old mond.
The new thing, let's call it NGT which is Moffat's term, does the same thing about explaining rotation curves without dark matter and handling the cosmological constant---and it connects with a version of DSR Smolin is working on with KowalskiGlikman--the TSR or triply special relativity socalled.
So there is some scuttlebut background from the May conference as well as the formal presentations. I am only guessing about the informal gossip but there is a lot of related stuff at Baez website now that came from people's response to his TWF 206

I want to try to put these things together and get some kind of picture to jell out of it---a picture of what is going on in Quantum Gravity in first half of 2004. A lot is

 

[marcus]

First thing is to follow the link Baez gave to his TWF 206 and read his account of what Smolin was talking about mond-wise, and then
read all the responses that Baez got about mond-ish stuff including critiques and a recent Bekenstein article.

but then look at a few scheduled talks
(not intended as a representative sample!)
-------------------------

Monday, May 3rd


J. Pullin (Consistent discretization)


------------------------------

Tuesday, May 4th

R. Loll (Dynamical triangulations)


---------------------------------------

Wednesday, May 5th


R. Gambini (Relational time in consistent discrete quantum gravity)


------------------

Friday, May 7th

J. Kowalski Gliksman : (DSR as a possible limit of quantum gravity)
F. Girelli (Special Relativity as a non commutative geometry: Lessons)

-----------------
Notice the merging of lines of research as they mature. DSR is not a theory of gravity it is just a modification of minkowski-space to make one more quantity invariant (besides c). but analogous to how old minkowski space was the tangent space or local streetmap for old GR, if we have a new quantum GR maybe it could have DSR as its local approximation. or
maybe an even better modification of minkowski space like TSR (that jerzy k-g and smolin are working on) or the DDSR that girelli and livine and oriti just posted on----so these Friday talks by Jerzy K-G and by Girelli are about that

and the other interesting thing about them is that they are not only merging DSR with QG, they are putting out feelers to Moffat's mond-ish Nonsymmetric Gravitational Theory (with its comprehension of dark matter and dark energy)-----because Girelli/Livine/Oriti said that explicitly in the paper they just posted, and they are working somewhat parallel with Smolin and JerzyKG and Smolin is talking about mondish stuff.

We arent going to have separate fields, it seems, because quantum gravity is making contact with and beginning to absorb things like DSR and MOND or versions or decendants of them.

And then it happened today that Jorge Pullin posted that paper on resolving the Black Hole Information puzzle---by Gambini and Porto (at Carnegie Mellon) and Pullin (at Louisiana)
I think it is an important paper because that puzzle has NOT till now been resolved, it is a real puzzle and GP and P are proposing a really simple solution.
And they were at the Marseille conference talking about relational time
and it is exactly thus they resolve the puzzle----absolute time is not real!
Absolute time does not exist in nature, all we have is whatever clocks we can manage to build or observe and they relate conditional quantum-fashion to other observables. OK they say, let us be realistic and use actual observable material clocks. Let us not pretend there is an absolute perfect clock that God winds up every day for all eternity, but only various imperfect clocks like your wife has.

then, Lo and Behold, the black hole information puzzle vanishes
(but there seems to be a nontrivial calculation to show this---two years ago they tried but didnt get it, then just now they got it)

with realistic (relational) time, evolution is just very slightly nonunitary!

(maybe our PF member called "Nonunitary" will like this)

and because of the very slight nonunitariness, information is not forever, it gradually fades out, but very very slowly

however black holes evaporate very slowly

so by the time the BH has evaporated all the information would have
faded into nonunitary oblivion ANYWAY
therefore no information is lost by the BH evaporating

Those friends and associates of Susskind who speculated about black holes leaving remanants or the information "teleporting" out of them by stringy business, they did not have to worry themselves about it. Theirs may merely have been a deluded effort to save perfect absolute-time unitarianism.

Why do I think Rovelli will be amused by Gambini Porto Pullin's paper
resolving the BH info paradox? Didnt he suspect already that understanding time better would do that?

 

[marcus]

the poet Borges said (and Wilbur, a great translator, translated)


"One thing does not exist: oblivion
God saves the metal and he saves the dross
and his prophetic memory guards from loss
the moons to come, and those of evenings gone..."

it is the first four lines of one of the most wonderful sonnets
ever written in english

but if relational time destroys the unitariness of time-evolution and
pure quantum states gradually lose coherence
and informations fades, even as dewdrops and black holes evaporate,
then Borges vision is incorrect.

he wouldn't have liked that, he more than any 20th century poet
tried to make sonnets and stories which were true to the general theory
of relativity and to quantum mechanics. especially his short stories which are true to quantum mechanics. he wanted his poetry to be correct.

 

[marcus]

I was typing from memory, here is a longer exerpt of Borges poem
the Letralia website has the complete poem in both languages
http://www.letralia.com/58/en02-058.htm

Everness

One thing does not exist: Oblivion.
God saves the metal and he saves the dross,
And his prophetic memory guards from loss
The moons to come, and those of evenings gone.
Everything is: the shadows in the glass.
Which, in between the day's two twilights, you
Have scattered by the thousands, or shall strew
Henceforward in the mirrors that you pass.
And everything is part of that diverse
Crystalline memory, the universe:
...

Everness

Sólo una cosa no hay. Es el olvido.
Dios, que salva el metal, salva la escoria
Y cifra en Su profética memoria
Las lunas que serán y las que han sido.
Ya todo está. Los miles de reflejos
Que entre los dos crepúsculos del día
Tu rostro fue dejando en los espejos
Y los que irá dejando todavía.
Y todo es una parte del diverso
Cristal de esa memoria, el universo;
...

Everything is: the shadows in the glass.

that is, no information is ever lost.

And everything is part of that diverse
Crystalline memory,

that is, 4D spacetime is a static eternity with all our worldlines
and the worldlines of all the particles which momentarily interweave to make us

 

[marcus]

...


Tuesday, May 4th

R. Loll (Dynamical triangulations)


---------------------------------------

Wednesday, May 5th


R. Gambini (Relational time in consistent discrete quantum gravity)


------------------

Friday, May 7th

J. Kowalski Gliksman : (DSR as a possible limit of quantum gravity)
F. Girelli (Special Relativity as a non commutative geometry: Lessons)

...

several of the talks at the Marseille symposium have subsequently appeared as papers

Girelli, Livine
"Special Relativity as a non-commutative geometry: Lessons for Deformed Special Relativity"
http://arxiv.org/gr-qc/0407098

Kowalski-Glikman, Smolin
"Triply Special Relativity"
http://arxiv.org/hep-th/0406279

Gambini, Porto, Pullin
"Realistic clocks, universal decoherence and the black hole information paradox"
http://arxiv.org/hep-th/0406260

BTW wasnt it great having Baez drop into PF and report from the Marseille conference, starting this thread!

I hope he makes it a habit. I would very much like to hear what he has to say about September's London conference in honor of Chris Isham.
Renate Loll will be one of the speakers.

 

[marcus]

moment of truth for simplex gravity (dynamical triangulations)

... I would very much like to hear what he has to say about September's London conference in honor of Chris Isham.
Renate Loll will be one of the speakers.

tomorrow the Isham 60th birthday conference at Blackett Lab Imperial College London

http://www.imperial.ac.uk/research/theory/about/isham60/schedule.htm

10AM tuesday is Renate Loll talk.

they posted a paper in April, computer study results,
"Emergence of a 4D world..."

Ambjorn Jurkiewicz Loll
"Emergence of a 4D World from Causal Quantum Gravity"
http://www.arXiv.org/abs/hep-th/0404156

and Renate presented the results in May at the Marseille conference.

It caused some stir because it seems there is some chance of real progress in that area. It came after about 15 years of people trying this approach with success only in lower dimensions.
Simplicial quantum gravity had seemed reasonable but had never generated a normal healthy 4D world in computer modeling, until the AJL paper.

that was May, now it is 4 months later, September. Has there been further progress or not?

John Baez is attending tomorrow's conference, but not giving a paper IIRC.
Maybe we will hear some word from him



The speakers include Hawking, Rovelli, Ashtekar, Penrose, Loll...

Here are some talks

K. Kuchar: Spacetime Covariance in Canonical Relativity.

J. Hartle: Arrows of Time and Generalized Quantum Theory

R. Penrose: What is Twistor-String Theory?

G. Gibbons: The First Law of Thermodynamics for Kerr-Anti-de-Sitter Black Holes in Arbitrary Dimensions

R. Loll: Emergence of a 4d World from Causal Path Integrals

S. Hawking: The Information Paradox for Black Holes

R. Sorkin: Is a Past Finite Order the Inner Basis of Spacetime?

C. Rovelli: How to Extract Physical Predictions from a Diffeomorphism Invariant Quantum Field Theory

A. Ashtekar: Recent Advances in Loop Quantum Gravity

 

[selfAdjoint]

Marcus, have you heard any more about this? Any of the talks posted online?

 

[marcus]

Marcus, have you heard any more about this? Any of the talks posted online?

I am glad that you are back sA,
I was expecting that John Baez, since he attended, would post something about it, but so far he didnt.

Maybe he would if we asked him nicely.

I am sorry to say that I have no lead on any of the London talks.

 

[marcus]

To a large extent the discussions in this thread were around the first AJL paper
Ambjorn Jurkiewicz Loll
"Emergence of a 4D World from Causal Quantum Gravity"
http://www.arXiv.org/abs/hep-th/0404156

and some of the "sidebar" material may be worth recalling.

Renate Loll presented the paper at the May conference.
We got some of John Baez perspective on it from him in this thread,
and in his TWF#206
and is parallel conversations with Larsson and others on SPR.

This thread has some links to some of that parallel discussion, and
also to an article about Simplicial Gravity---or Dynamical Triangulations---
that Matt Visser had in Jorge Pullin's newsletter Matters of Gravity

 

[marcus]

To a large extent the discussions in this thread were around the first AJL paper
Ambjorn Jurkiewicz Loll
"Emergence of a 4D World from Causal Quantum Gravity"
http://www.arXiv.org/abs/hep-th/0404156

John Baez introduced the Dynamical Trianglulation (DT) quantum gravity approach to us at PF by starting this thread and highlighting the above paper in his report from the May 2004 conference.

I guess this is our main DT thread. I'm going to do an "introduction to DT"
here. I will put links to tributary threads, and to biblio.

For me, after the April 2004 paper, there was a waiting period to see how things would go. I think DT now looks stronger than ever, as a proposed QG.

the best introduction to DT that I have been able to find in the literature is
sections of a 96 page paper by AJL, posted January 2000
LORENTZIAN AND EUCLIDEAN QUANTUM GRAVITY– ANALYTICAL AND NUMERICAL RESULTS
http://arxiv.org/hep-th/0001124

this is the closest thing to the introductory chapters of a textbook, as yet. but it has non-essential sections that deal with problems they were having back in 1999 and 2000.

there are some later AJL papers that carry on the introductory exposition,
after this one. I want to map out how to piece together a kind of beginning text.

BTW I think DT is turning out to be a serious rival to any quantum gravity theory you can name. thanks to John Baez for alerting us to it.

 

[marcus]

some PF threads discussing DT

https://www.physicsforums.com/showthread.php?t=54262

https://www.physicsforums.com/showthread.php?t=53974

https://www.physicsforums.com/showthread.php?t=52958
(I didnt do the greatest job explaining in that thread,
and want to remedy that and give a better account)

https://www.physicsforums.com/showthread.php?t=55806
(this has some bibliography)

 

[marcus]

the history of DT

DT is based a modified version of Regge calculus
So it goes back originally to Tullio Regge's landmark 1961 paper
General Relativity without Coordinates
which showed how to do a discrete Einstein equation
in a triangulated 4D space (a space divided up into 4simplexes)

Regge's method involved knowing the lengths of the edges of the simplices and doing arithmetic with them. He could get a substitute for curvature without ever taking the derivative.

the first distinctively DT approach was around 1985 in 3 separate papers:
Ambjorn et al, by F.David, and by V.A. Kazakov, I.K. Kostov and A.A. Migdal.

What made DT different was you made all the 4simplexes be identical, or all of a small number of types. Then all that matters is COUNTING. counting numbers of simplexes, and vertices, and edges etc.

that is, DT is different from Regge style because Regge allowed for individual variation in the size and shape of simplexes, so everything depended on measuring the individual simplexes in some locale. but
DT just uses some stock simplexes and counts. But it also works.

So starting around 1985, Ambjorn et al got into trying to do quantum gravity with DT.

Particularly they wanted to do a path integral approach, the idea of which had been made popular by Stephen Hawking. And they started doing Monte Carlo computer runs with random 4D triangulations (and lower dimensional analogs) to evaluate the path integral.

DT suffered from a lot trouble and the random triangulated spacetimes were always crumpled or fractal-feathery, or plagued by budded-off "baby" universes. So for over 10 years it seemed discouraging.

It seems to have been around 1998 that Ambjorn and Loll got the notion of restricting DT to a kind of FOLIATED triangulation which would have some causal or Lorentzian structure.

they began a program of working up from 2D to 3D to 4D
and it worked at each stage and got better all the time
and this finally led to the two papers that posted this year.
which kind of put this approach on the map

 

[selfAdjoint]

Marcus, I want to thank you for going through all this and keeping us up to date, and particularly the fine explanations you have worked up about DT. As your discussion of Oriti's latest paper suggests, this DT program may be about to converge with other approaches to quantum gravity - sort of the way K-Mart merged with Sears, where the stores will all become Sears named but the management will all be K-Mart.

 

[marcus]

Nightcleaner,
the Letralia website has the complete poem in both languages
http://www.letralia.com/58/en02-058.htm
and many more besides this one

Everness

One thing does not exist: Oblivion.
God saves the metal and he saves the dross,
And his prophetic memory guards from loss
The moons to come, and those of evenings gone.
Everything is: the shadows in the glass.
Which, in between the day's two twilights, you
Have scattered by the thousands, or shall strew
Henceforward in the mirrors that you pass.
And everything is part of that diverse
Crystalline memory, the universe:
Whoever though its endless mazes wanders
Hears door on door click shut behind his stride,
And only from the sunset's farther side
Shall view at last the Archetypes and Splendors.


Everness

Sólo una cosa no hay. Es el olvido.
Dios, que salva el metal, salva la escoria
Y cifra en Su profética memoria
Las lunas que serán y las que han sido.
Ya todo está. Los miles de reflejos
Que entre los dos crepúsculos del día
Tu rostro fue dejando en los espejos
Y los que irá dejando todavía.
Y todo es una parte del diverso
Cristal de esa memoria, el universo;
No tienen fin sus arduos corredores
Y las puertas se cierran a tu paso;
Sólo del otro lado del ocaso
Verás los Arquetipos y Esplendores.

...

here is one that Letralia doesn't have:

to see a world in a grain of sand
and a heaven in a wild flower,
hold infinity in the palm of your hand,
and eternity in an hour

I don't know what you are talking about
I know what you are talking about

pick one

 

[marcus]

sorry everybody
I got off topic
it is probably better to start a separate thread for poetry et al. and
let this one stay focused on what John Baez called attention to:
the AJL paper
Dynamical Triangulations

 

[selfAdjoint]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Go back to Marcus' earlier post about Regge Calculus. Years ago Tullio Regge triangulated GR spacetime and by doing combinatorial things with the edge-lengths of the triangulation he was able to do all the GR curvature math that is usually done with tensors and differential forms and second derivatives. Then Ambjorn and coworkers made all the lengths the same size and revised the combinatorial shuffle to an even simpler form, but they had problems and it was a years-long slog to get to their present causal triangulations which work so splendedly.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

 

[marcus]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Go back to Marcus' earlier post about Regge Calculus. Years ago Tullio Regge triangulated GR spacetime and by doing combinatorial things with the edge-lengths of the triangulation he was able to do all the GR curvature math that is usually done with tensors and differential forms and second derivatives. Then Ambjorn and coworkers made all the lengths the same size and revised the combinatorial shuffle to an even simpler form, but they had problems and it was a years-long slog to get to their present causal triangulations which work so splendidly.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

classic epigrammatical account, wanted to email it to Ambjorn as a kind of
maximally concise statement of their work's place in the q.g. story.
won't though, since they must have plenty to think about without
e-fanmail

 

[marcus]

photo of Tullio Regge

http://www.icra.it/MG/awards/Images/regge.jpg

born 1931 Torino
discovered Regge calculus 1961 while at Princeton
where he worked with John Archibald Wheeler

 

[marcus]

I still think the best detailed introduction to AJL dynamical triangulations
is "Dynamically Triangulating Lorentzian Quantum Gravity"
http://arxiv.org/hep-th/0105267

more than one person at PF has indicated they'd found it useful,
printed it out, etc. Also AJL refer back to it as a basic reference
several times in their recent (2004) papers.

would also be nice to have an online source giving an
introduction to Regge calculus----hopefully would have pictures
since the subject could be presented visually

for basic path integral terminology, here is the Wiki article
on "path integral"
http://en.wikipedia.org/wiki/Path_integral_formulation

if you wish, this will lead you back to contributory Wikis on "action", "Lagrangian" etc.

Here's a brief introduction to Regge calculus (esp. as applied to numerical relativity) by Adrian Gentle
http://arxiv.org/abs/gr-qc/0408006
it really has barely a page actually explaining R's discrete gen. rel.

hope we find more. I will keep looking

 

[nightcleaner]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

Yes, in GR spacetime is one unified thing, but my reading has led me to think that GR isn't used to describe the world at the Planck scale, but is considered to be a poor model of events in the very small, very high energy realm. GR is a cosmological paragigm, while the standard model of particles in flat space is more often used in discussions of the very small. Did I miss something?

Now, it is very nice of you to invite me to try to match the work of PH.D's at two major European universities and The Max Planck research institute, backed up by all the departmental machinery and academic freedoms they have available to them, while I am nothing but a nightcleaner in a tourist restaurant. Actually I am gratified by the fact that AJL has done work in the very field I have been unsuccessfully trying to draw attention to here and in previous years on SST.com.

I feel somewhat as a bean farmer must who finds a fertile plot of ground, scratches at it with a stick and makes a little crop for a few years, then finds himself and his tender garden uprooted by the massive machinerey of agribusiness. It seems they want to build a driveway for their factories on top of my little digs, and I may as well get out of the way or get paved over. Huh.

Well, it is no real surprise. And I have the small satisfaction of saying that I, at least, knew where to dig. And I take away something else as well. I may not be able to apply the Regge calculus (at least not yet) but my model is prettier.

nc

 

[marcus]

... I may not be able to apply the Regge calculus (at least not yet) ...

hello NC, i am still groping around for introductory material on Regge calculus and the closely related DT approach.

here are some online page references, if for no other use than my own!
I found parts of these helpful.

Loll 98----pages 8-13 are about standard Regge
pages 14-17 are about DT
http://arxiv.org/gr-qc/9805049
Discrete approaches to quantum gravity in four dimensions
this is a "LivingReviews" survey article that the AEI invited Loll to contribute
it surveys current (1998) research in several related areas and give
a large bibliography. She includes a halfdozen or so introductory sources on Regge calculus but none are online. her own treatment is quite concise.


For a more elemenary discussion: try Loll 02, pages 8-16
see also the summary at the end pages 34 and 35.
http://arxiv.org/hep-th/0212340
A Discrete History of the Lorentzian Path Integral

this is a pedagogical article, to help get graduate students involved.
It is historical, describing difficulties as they were encountered. I find this often helps me understand.
this essay is willing to waste words explaining some simpler points that a normal research article would not explain

However the rest of the article----pages 1-7 and 17-33
is much concerned with the problems that were being encountered in 2002!
since they have gotten past some of that, it is less interesting now IMO.

Maybe as a sample I will quote some from around page 8.

 

[marcus]

Here is a sample from around page 8 of Loll 20 survey
http://arxiv.org/hep-th/0212340
A Discrete History of the Lorentzian Path Integral

this is just to give the flavor. I will not bother to reproduce the math symbols exactly but will simply drop symbols in some cases---leaving whatever copies easily: the words.

---sample---
“Lorentzian dynamical triangulations”, first proposed in [13] and further elaborated in [14, 15] tries to establish a logical connection between the fact that non-perturbative path integrals were constructed for Euclidean instead of Lorentzian geometries and their apparent failure to lead to an interesting continuum theory. Is it conceivable that we can kill two birds with one stone, ie. cure the problem of degenerate quantum geometry by taking a path integral over geometries with a physical, Lorentzian signature? Remarkably, this is indeed what happens in the quantum gravity theories in d < 4 which have already been studied extensively. The way in which Lorentzian dynamical triangulations overcome the problems mentioned above is the subject of the Sec. 5.

4 Geometry from simplices
The use of simplicial methods in general relativity goes back to the pioneering work of Regge [16]. In classical applications one tries to approximate a classical space-time geometry by a triangulation, that is, a piecewise linear space obtained by gluing together flat simplicial building blocks, which in dimension d are d-dimensional generalizations of triangles. By “flat” I mean that they are isometric to a subspace of d-dimensional Euclidean or Minkowski space. We will only be interested in gluings leading to genuine manifolds, which therefore look locally like an R^d. A nice feature of such simplicial manifolds is that their geometric properties are completely described by the discrete set ... of the squared lengths of their edges. Note that this amounts to a description of geometry without the use of coordinates. There is nothing to prevent us from reintroducing coordinate patches covering the piecewise linear manifold, for example, on each individual simplex, with suitable transition functions between patches. In such a coordinate system the metric tensor will then assume a definite form. However, for the purposes of formulating the path integral we will not be interested in doing this, but rather work with the edge lengths, which constitute a direct, regularized parametrization of the space Geom(M) of geometries. How precisely is the intrinsic geometry of a simplicial space, most importantly, its curvature, encoded in its edge lengths? A useful example to keep in mind is the case of dimension two, which can easily be visualized. A 2d piecewise linear space is a triangulation, and its scalar curvature R(x) coincides with the so-called Gaussian curvature. One way of measuring this curvature is by parallel-transporting a vector around closed curves in the manifold. In our piecewise-flat manifold such a vector will always return to its original orientation unless it has surrounded lattice vertices v at which the surrounding angles did not add up to 2[pi], but [formula omitted]
see Fig.4. The so-called deficit angle [delta] is precisely the rotation angle picked up by the vector and is a direct measure for the scalar curvature at the vertex. The operational description to obtain the scalar curvature in higher dimensions is very similar, one basically has to sum in each point over the Gaussian curvatures of all two-dimensional submanifolds. This explains why in Regge calculus the curvature part of the Einstein action is given by a sum over building blocks of dimension (d-2) which are simply the objects dual to those local 2d submanifolds
---end quote---

Notice how new Causal DT (Lorentzian DT) is! She says it was first proposed only in 1998-----the reference [13] is to a paper by Ambjorn
and her.

I have bolded "...which constitute a direct, regularized parametrization of the space Geom(M) of geometries..."

you have a formless continuum M, and you make a "space" consisting of all the possible geometries you could have on M. this is where the quantum state of the geometry is going to live, or be defined. In another of her writings Loll calls this space of geometries, this Geom(M) the "mother of all spaces" or something like that.

this is a more direct "quantization-ready" approach than some others (e.g. think of the Ashtekar variables). In the straight Regge, there is just this long list of EDGE LENGTHS and that effectively describes a geometry and coordinatizes Geom(M)

now DT insists that all the edges are standard lengths and so instead of a list of edgelengths you have a list of what is next to what, recording the "connectivity"---it should be simple enough: some ways of writing it down would be more efficient than others----some computer data structure that names the vertices and says which ones are vertices of what tetrahedron etc., enough information so you can tell what is a side of what.

 

[nightcleaner]

hello NC, i am still groping around for introductory material on Regge calculus and the closely related DT approach.

here are some online page references, if for no other use than my own!
I found parts of these helpful.

...

Hi Marcus, and thanks for posting your finds here. I do think they are helpful.

However, I have trouble reading the math, and the papers are full of jargon which make them difficult for ordinary English speakers such as myself. I have been reading physics for a couple years and trying to improve my math skills, so I think I have some idea of what AJL are trying to convey. Still, I find myself taking a drubbing on the forehead when trying to read Loll and her collegues.

Are you willing to entertain questions on the math and physics here?

For example, here is a web link from Mathworld. It seems to be relevant, but I am not sure, and will withdraw it from the forum if it is not to the point of this thread.

http://mathworld.wolfram.com/Simplex.html

nc

3790